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Wednesday, June 27, 2012

Nature = Mathematics?

On the weekend I had to spend some time at the airport. Something about airports has me come back to the question just what is reality anyway. In this case it was a long hallway down the terminal that sparked the thought; it had me think about perspective drawing.

I taught myself perspective drawing in 5th grade, which I recall because my friends asked me to explain how to do it, upon which I went to the library to learn it proper. I was surprised to read how late in the history of painting it was that artists got the perspective right. Upon closer introspection I guess though I didn’t actually learn it from watching the three dimensional world carefully, but by watching carefully images and photos that were already two dimensional.

There are some early examples of perspective in drawing, it was for example widely known that objects in the distance appear smaller, but it wasn’t until the 15th century that the geometrical methods were properly developed and widely used. I don’t think it’s a coincidence that this was briefly before the scientific revolution dramatically changed the way people understood the world.

A painting is, in a very simple sense, a model of the world, and understanding perspective drawing must have had people realize that there is a mathematical basis of the world that’s waiting to be recognized. If you make an accurate drawing of, say, what you see out of your window, if you have sufficient details about how the mountains look like and where the river is, you might be able to “predict” from your drawing that there must be a tree standing over there.

I used to think of our theories as being maps, essentially, from mathematics to reality. I wrote about this earlier and will just reproduce here the accompanying diagram.

There is the world of mathematics, the eternal platonic ideal, and we take a part of it and identify it with the real world. The mathematical part we can call “the model” and “the theory” is the mechanism of identification with the real world, essentially how you compute observables and connect them to data. (I am aware that’s not how other people might be using these words, but arguing about words is pointless.)

This picture of the way we describe the world however raises the question if there is a distinction between these two areas, the question whether mathematics is equally real as that computer screen you are looking at, a question that is some thousand years old, minus the computer. Note that to ask that question, I don’t have to tell you what “real” means. I am just asking if there is a difference between a mathematical object and something that you can throw at me. Max Tegmark famously does not believe there is a distinction.
Most people I know believe there is.

However, it occurred to me, the mapping that the image suggests is actually not what we do if we build a model or apply a theory. What we always do, instead, is that we map one system of the real world to some other system, where the idea is that the one system is better to understand or to use.

Think of the painting: the painting is not a mathematical object. It’s an abstraction, all right, one that can make use of mathematical tools, but it’s not in and by itself a platonic idea. The same is true for all other models that we use. A computer simulation is not a mathematical object, it is a re-building, usually also a simplification, of another part of nature that we want to compare it too. And a calculation that you do in your head is not platonic either, it’s some firing of neurons and a lot of chemical reactions going on, and so on. And it is again, essentially some simulation, approximation, extrapolation, of another part of nature that you want to compare it to, to the end of making a prediction because you want to know if you got it right.

So where does that leave mathematics then? Mathematics is a tool that we use to improve on our models, it’s a technique that we force our thoughts through because it has proven to be incredibly useful. Nevertheless, the point I am trying to make is that this usefulness doesn’t mean a model actually extracts some mathematical “substance” from reality.

You will wonder now what does it matter. The reason it matters to me is that for reasons I elaborated on in this earlier post, I think that the occurrence of the multiverse in its various forms is unavoidable and a consequence of relying exclusively on mathematical consistency. The multiverse tells us that mathematics is not sufficient. What is, I don’t know.

The question is of course if we can conceive of any type of model and a theory to map it that is not mathematical. One thing that came to my mind here is analog gravity, basically the idea to study some types of gravitational phenomena with condensed matter or fluid analogies (thus the name), an idea that has caught on during the last years. I am not terribly excited about this because I don’t really see what we learn from this about quantum gravity. But the point is that it’s an example where you have a model (the “analogue”) that is mapped to the system you want to describe (spacetime) and the model in this case is not a mathematical structure.

Or in other words, if it should be the case that nature cannot be described by mathematics alone, this type of models could still be used.

So much about my latest thoughts on the question whether, at some point in the history of science, we will have to find a way to go beyond mathematics to make progress, and what that could possibly mean.

To them, I said,the truth would be literally nothingbut the shadows of the images. -Plato, The Republic (Book VII)

Also consider this process as an examination toward expression and maybe this will provide for some thoughts about the dimensional significance of how one may look at the expression of mattered patterns that exist in nature.

The basis of holography and matter expressions all have their respective features in how one sees "the shadows in the cave."

Of course, to Plato this story was just meant to symbolize mankind's struggle to reach enlightenment and understanding through reasoning and open-mindedness. We are all initially prisoners and the tangible world is our cave. Just as some prisoners may escape out into the sun, so may some people amass knowledge and ascend into the light of true reality.

What is equally interesting is the literal interpretation of Plato's tale: The idea that reality could be represented completely as `shadows' on the wallsSee:The Holographic Principle and M-theory

This is a orientation of sorts in terms of perspective.

The artistic rendition has it's corollary in how the cubists may have seen" abstraction model perspectives" with regard to how one may look at reality. They sought to capture this. Dali in his idea of the crucifixion as an explanation of what he saw as what heaven may be interpreted as?

Hi Bee,This is a very important and provocative subject you have brought up. I also think you have the correct slant on it. I think what you are saying is that the correct philosophy of science is required as an underpinning for practical science to move forward when old paradigms outgrow their usefulness.

Mathematics is just the tool that correctly shows how to apply science to specific physical problems. I think it will always be essential, especially in physics. The problem is that many physicist today have gone off the rails in not understanding that physical understanding is a subset of all mathematical knowledge. There is far more mathematics than there is models that describe our physical world. In other words, accurate mathematical models of the universe is a small subset of all mathematical models available.

What this means os that when a roadblock occurs in science, specifically physics, you must turn from the technicians most proficient in math to people who are better at visualizing. It also implies that the universe must be constrained in it's laws to have any wide applicability. Otherwise it would not even be possible to visualize the laws of the universe.

So perhaps the most practical philosophical signpost to the direction forward comes when a large population of scientists, both amateur and professional, can actually visualize in their mind how the universe might be working. This visualization must take into account the constrained quality of the universe in which any change in one quantity will necessarily affect the quantity of something related it to that can actually be observed. These observations can be both local and non-local, but the essential quantity, (I prefer to call it the duals, time and energy) is always conserved and constrained.

The mathematical modeling comes after enough people can reach a consensus on how what we see and visualize is manifested in the physical world.

Another simpler way of saying how things got so screwed up today is that mathematical physicists, like Max Tegmark, are acting like little kids. Since the only tool they have in their arsenal is math - the hammer- they treat the universe as a nail to be hammered with it. They are all like a bunch of little kids hammering away and making a lot of noise and exclaiming to each other "Look what I made!"

Ah - so that is Tegmark's article. Thanks. It sounds like he identifies the physical reality with a mathematical description, like a Theory (of everything) - expecting a collection of equations. Then he says it must predict a multiverse because it must be a complete description of reality. What bothers me is that it ought to make sense of why "particles" exist in the first place - which seems prior to describing what they do. But perhaps the Mathematical Object is more like a Definition than a Description.

Ah - so that is Tegmark's article. Thanks. It sounds like he identifies the physical reality with a mathematical description, like a Theory (of everything) - expecting a collection of equations. Then he says it must predict a multiverse because it must be a complete description of reality. What bothers me is that it ought to make sense of why "particles" exist in the first place - which seems prior to describing what they do. But perhaps the Mathematical Object is more like a Definition than a Description.

Isn't a long lesson out of the history of mathematical physics, that mathematics gives a whole lot of unphysical solutions which have to be pruned out of the physical theory? Why does a multiverse that arises from the mathematics necessarily have to be accepted?

"The main point of this presentation is that it is far from obvious what vision is for – and J.J. Gibson’s main achievement is drawing attention to some of the functions that other researchers had ignored."

and

"But that common idea is mistaken: visual systems do not represent information about 3-D structure in a 3-D model (information structure isomorphic with things represented)but in a collection of information fragments, all giving partial information.A model cannot be inconsistent.A collection of partial descriptions can."

The partial descriptions which are inconsistent by definition cannot arise from one mathematical "reality".

This is a deep issue indeed. In essence, even the platonic mathematical world is yet to be fully explored. For example, in the quest to complete a quantum theory of gravity, it's expected that new mathematics will be invoked. Yet who will create such mathematics? It could very well be a physicist who fashions the needed mathematical tools, as Newton once did. So in this case, nature serves as a guide through the uncharted platonic-mathematical wilderness. Our universe very well may be a subset of a higher mathematical construct, but without studying nature it is difficult to even grasp the complexity of such a construct.

But the model systems for studying gravity are only useful because they have mathematical similarities to gravity. That is, a priori we have a mathematical abstraction of both systems, and since the mathematics is the same, we can test our understanding of that mathematics by making predictions and asking Nature what It really does via experiment. So to me it still boils down to mathematics, and the observations of the model systems are only accurate representations of gravitational phenomena insofar as we can map the mathematical theories onto each other.

I fail to see how it is useful to think of the equations we scrawl on blackboards as a literally physical model for some other physical phenomenon. To me the stumbling point is that fact that the last equation you write will have followed from the first (or however many given equations you need) whether or not you did the physical act of writing it. The mathematics is all there - all contained in itself - independently of its expression in a physical medium.

I think one needs to take more seriously the ontological status of mathematics than is implied in this blog post. That doesn't mean we live in a mathematical universe. It means considering questions like: What does it mean for a theorem to be "true?" How is that different from/the same as what it means for something to be true about Nature?

On the other hand, I think your viewpoint can be made much stronger by making the statement that mathematics IS the mapping of natural (i.e. physical) patterns onto one another, which to me explains the efficacy of mathematics in describing nature. Nature follows patterns too, and sometimes the mathematical patterns we humans create happen to have things in common with the patterns of nature. Many created patterns do not (or at least do not obviously appear to, although we could always find some instantiation of them later). I find this view to be quite satisfying, actually.

RE: "The question is of course if we can conceive of any type of model and a theory to map it that is not mathematical."

A. Life evolves via genetic mutations and survival of the fittest.

B. Ontogeny recapitulates phylogeny.

C. The Sun is a star.

D. Matter is composed of atoms and subatomic particles.

1. In the case of biological evolution, mathematics takes a back seat compared to conceptual analysis.

2. Many profound things about nature can be best expressed conceptually.

3. Pictoral and graphical modeling using computers might one day conceivably provide arbitrarily accurate models of natural phenomena. Math would clearly be involved, but numerical modeling is a different use of mathematics, with different expectations and assumptions about the role of mathematics.

One is communication. The point is to tell someone else what you saw, where you went, what someone did, how so and so felt and so on. Depending on societal norms, different things were considered more or less important. The ancient Greeks knew a fair bit about perspective. They used an inverse transform to make temple pillars look straight rather than tapered. They just didn't use it when drawing representative scenes because it didn't communicate anything of importance.

The other goal is prediction. It turns out that it is possible describe certain things symbolically and then manipulate the symbols in a rigorous manner to understand the past, the future or to find analogies. If you only adopt representations and rules of manipulation that work, then you have a mathematics or logic or method of reasoning or whatever you want to call it.

There may be a universe where it is impossible to reason forward or backward in time or to find analogies, but that universe would lack thought.

You shouldn't rely on mathematical consistency only. You could find a physical selection principle to narrow down the possibilities. Maybe there is no such principle but in any case you can't blame mathematics for this. Similarly you can't blame mathematics because there are many string vacua you should blame the physical theory. Moreover Quantum tunneling or eternal inflation are physical not mathematical concepts.

Yes, you are right. This one sentence is a very brief (too brief) summary of my longer post that I linked to. You need some other addition than mathematical consistency for your theory, for example tying it to observation, yes. But fundamentally, what it tells you is that pure mathematics isn't enough one way or the other. Best,

/*.. the point I am trying to make is that this usefulness doesn’t mean a model actually extracts some mathematical “substance” from reality. */

You probably wanted to say, the formal numeric regression of reality often doesn't reveal its physical nature at all. For example, what the Newton's gravitational law actually says about physical mechanism of gravity? IMO the ancient LeSage model is way more consequential in it.

/* The multiverse tells us that mathematics is not sufficient. What is, I don’t know. */

IMO the multiverse concept is so poorly defined, it doesn't say anything. But IMO the math is not hopeless with it. AWT has two dual formulation: the nested dense particle fluctuations and the waves spreading in infinitely many dimensions. The later approach is solely formal and IMO it's equivalent to the first model. Who actually attempted to model the waves in sufficiently high number of dimensions? Maybe we would be surprised, what this model is actually capable of...

“Is there a difference between a mathematical object and something you can throw at me?”

I think the answer to this can only be decided if its determined if you are nothing more than a mathematical object or not. More seriously I think it ultimately comes down to whether the essence of objects themselves are more than simply place markers, where mathematics serves to describe the positions they hold, how they are ordered and the how the evolution of this ordering proceeds. For me I’ve always thought this essence needs to be necessarily more than this, as how otherwise what is there to distinguish the platonic world from what we recognize to be reality. So perhaps you are right to feel that it’s not enough for physical theories to look only to the real, yet further beyond to what some might consider to be the surreal.

“Surrealism is destructive, but it destroys only what it considers to be shackles limiting our vision.”

"... if it should be the case that nature cannot be described by mathematics alone ..." Physics needs to model the psychology of measurement and the foundational physics of measurement. In terms of the foundations of physics, I would bet on either: (1) M-theory with the infinite nature hypothesis and the string landscape or (2) modified M-theory with Wolfram's mobile automaton using the monster group and the 6 pariah groups to derive approximations to the Lie groups of quantum field theory. The psychology of measurement might be somewhat inaccessible to mathematical analysis.

Let us say that formal mathematical reasoning began around the time of Euclid, some 2500 years ago. Let us say that science as we understand it today began around the time of Newton, or some 300 years ago.

Each maybe with someone else or sometime a bit earlier. But the point I'm trying to make is the relative recentness of the appearance of these methods of acquiring knowledge compared to the long lineage of man - even if you take humans to have the potential mental capability to acquire and use these methods even for only the last 10,000 years, instead of the hundred-fold longer period of the million+ years of human evolution.

Much before the appearance of the formal mathematical method, and the scientific method, could anyone have dreamed of these methods and their effectiveness?

Can we imagine additional effective methods of acquiring knowledge? Is our inability to imagine them a proof that there can be no such methods? It would be somewhat arrogant to imagine that we've exhausted the possibilities so soon.

Or are we at the dawn of a third method, that vaguely strikes our intuition? The new cognition enhancing tools we have are the computer and the network, and maybe collaborative mathematics and science enabled by the web are just our first fumbling steps towards what we cannot yet grasp in our imagination.

I don't see what the 'psychology of measurement' matters. A machine can do a measurement just as well and who cares, fundamentally, if it is eventually read by a human or an alien or by nobody at all. Or, the other way round, the brain is just another measurement apparatus. Best,

Around 1956, Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's works Circle Limit I–IV demonstrate this concept. In 1995, Coxeter verified that Escher had achieved mathematical perfection in his etchings in a published paper. Coxeter wrote, "Escher got it absolutely right to the millimeter."

I don't expect that if you never gone inward, one might never had identified some aspect of consciousness as peering ultimately to the very fabric of nature itself? Saw it's mathematics?

I mean.....you want to describe this journey in some way? Is it mathematical?

Bee, what you seem to be driving at is what Bohr already recognized a long time ago: "There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature..."

In other words, physics is a means of representing reality, not a direct apprehension of it. As a means of representation it belongs to semiotics as much as to physical science. This is I think a major stumbling block to physicists who have focused too narrowly on science as traditionally understood and not made enough of an effort to broaden their horizons into fields such as semiotics, phenomenology, philosophy, etc.

Yes, I have great sympathy for Bohr and I share his attitude that science is in the first line about describing nature. That alone however doesn't address the question how to best do it, which has me wondering what the role of mathematics is, or is not. Best,

Plato's cave is well rooted in the understanding of quasicyrstals? If you understand Laue's experiment then you are understanding what is happening on the screen(2D surface)? Possibly why tessellations are important from "another perspective." You are building a foundation. These are expressive from "a point" and lead into the future of projective geometries? Is it mathematically consistent as a basis of foundations in relativity?

OK, RLO, first of all Bohr was most certainly not a Platonist. Plato was the one who brought up the cave analogy and argued that "true" reality was elsewhere, at some more fundamental level than simply the shadows we see cast on a wall. In other words, Plato was the one arguing for hidden variables, aka the famous Platonic ideas, NOT Bohr.

Bohr was the one who argued that, at least as far as quantum physics is concerned, all we have are the shadows, and experiments that can be conducted using them, and that there is no more fundamental reality (i.e., Platonic idea) elsewhere to cast any shadows. In other words, "there is no quantum world."

You might not agree, but this is most certainly NOT a Platonic view, in fact just the opposite.

Secondly, if you are reluctant to get into semiotics in any detail, I completely sympathize. It's become and absurdly convoluted enterprise. What I would argue for is an awareness of some of the basic principles of semiotics, not necessarily the study of this field in any detail.

There is no question that science is a method of representing reality (assuming reality exists in the first place) and not to take this aspect into account, or moreover to discount the importance of the philosophical principles that ground both science and semiotics, is to condemn oneself to a life of confusion and futility.

Bee, you seem to be thinking in terms of a type of perspective or mathematics that gets closer and closer to reality as more and more detail is provided, or as increasingly effective methods are employed. So regardless of the present drawbacks, it can always improve and we can thus come closer and closer to reality, just as in calculus one gets closer and closer to the limit.

All I can say, as someone who has spent a lot of time with the history and theory of art, is that there is a fundamental problem with perspective representations that first became apparent in the work of Cezanne. Not many people realize that he was a realist at heart. So were the cubists, in fact, at least they started out that way.

What they discovered was that the closer you get to your subject, i.e., the more intensely you examine it and try to represent it, the stranger it becomes. And this appears also to the be the case with physics. In other words, Bohr's notion of complementarity comes into play in the visual arts as well as science. Which means it is not fundamentally a problem in physics or science, but a problem with representation itself.

Which is why it them morphs into a problem in semiotics. I've published on this issue and if you're willing to slog through it (not easy going I admit) you're invited to take a look at an essay of mine entitled "Passage from Realism to Cubism." http://doktorgee.worldzonepro.com/passage.htm

You can skip over the overly academic introduction. The meat begins with the section entitled: To See a Sight.

DocG: "OK, RLO, first of all Bohr was most certainly not a Platonist. Plato was the one who brought up the cave analogy and argued that "true" reality was elsewhere, at some more fundamental level than simply the shadows we see cast on a wall. In other words, Plato was the one arguing for hidden variables, aka the famous Platonic ideas, NOT Bohr."

The concept of the orbital differs from Bohr's concept of the orbit. Bohr considered an orbit to be a path that the electron always followed much like a train stays on a track. The concept of the orbital was developed in Schrodinger's work to avoid violating the Heisenberg Uncertainty Principle. In the Modern Theory of Atomic Structures a picture of an orbital is also called a Probability Diagram. By agreement among chemists, the orbital is a 90% Probability Diagram. This idea allows the electron to be found anywhere and still indicates where the electron spends most of its time.

You do not mess with RLO's orbitals and their cosmological associations:)

The star Eta Carina is ejecting a pair of huge lobes that form a "propeller" shape. His counting has to start some place as well? A starting point then is the issue?

Well, I might answer: introduce the concept of "one" and then you are off and counting.

But any "beginning" or "starting point" in nature is an entirely arbitrary and subjective human hang-up. There is no absolute space, time or scale in nature, only very limited quasi-"absolute scale" within individual cosmological Scales [...Atomic, Stellar, Galactic,...].

What "starting point" do you refer to, he asks half expecting a coherent answer?

More precisely, mathematics deals with computable ideas from the Platonic realm of ideas. These are ideas which can be represented completely by a finite sequence of digits, see Tegmark's paper "The Mathematical Universe". However, even in math there are non-computable objects, like non-computable irrational numbers (the constructivists deny their existence).

Another example of a non-computable idea is the idea of passage of time, and some quantum gravity researchers believe that this is an illusion, i.e. that time can be reducedto a computable idea, which I do not believe. Although the rate of passage of time depends on the observer's trajectory, I beleive that there is a global cosmic passage of time that is fundamental and cannot be stopped or reversed. I also beleive that human mind is essentially a non-computable idea (you may be able to describe some aspects of the human mind by mathematics, but it can be never described completely by it - as argued by Penrose, the reason are Goedel's theorems).

It's easy to see why time cannot be reversed. Imagine a sequence of events and then imagine the same events in reverse order. What you are imagining in the second case is a series of events moving forward in time, but in reverse order. If time itself were reversed then such a reversal would be imperceptible, because there would be nothing to measure it against.

Thus,any attempt to posit time reversal automatically puts into play a metatime which must still be understood as moving "forward." If you want to reverse this metatime, then you will need to posit a meta-meta time to measure it against, and so on ad infinitum.

What is more interesting is the question of whether time moves from the present to the past or from the past to the present. What begins in the present winds up in the past, so one would assume the present comes first. But the past always precedes the present, doesn't it, so it would seem that the past comes first.

Considerations of this sort lead to the realization that time, like space, is fundamentally a gestalt phenomenon. Time in this sense is the ground against which events, the "figures," take place. Events happen, things change, things move. But time in this sense appears as an immovable background. And just as we can have negative space, in which the ground comes forward, we can also have negative time, in which the temporal ground comes forward -- as stasis.

As you can see, time is riddled with paradoxes. And I don't think there is any way to formulate a test that would resolve any of the paradoxes. Which means that time, which Kant called a Transcendental Aesthetic, lies beyond the realm of physics and can only be understood, if at all, metaphysically.

""I don’t think it’s a coincidence that this was briefly before the scientific revolution dramatically changed the way people understood the world.""

Hmmm, it is an irony, that the example you show from Perugino shows the foundational propaganda lie of the catholic shurch: Jesus handing the keys (of heaven) to St Peter. This is basis for selling indulgence for money. In the background You see what the money was "needed" for: St. Peter cathedral, then only a partly ruined church from Konstantines time. Georg

Doc G, You are almost there. We experience time as a series of events and so science treats it as a measure from one event to another, so it's past to future, but as the changing configuration of what is, it's the future becoming the past. Ask yourself if the earth travels the fourth dimension from yesterday to tomorrow, or does tomorrow become yesterday because the earth rotates? The difference is that with the former, time is some foundational geometry of reality, but with the latter, it's a measure of action, similar to temperature. I just wrote an entry in the FQXi contest on this topic, so rather than clutter up this post, I'll link to that:http://www.fqxi.org/community/forum/topic/1304

“... is the relative recentness of the appearance of these methods of acquiring knowledge compared to the long lineage of man - even if you take humans to have the potential mental capability to acquire and use these methods even for only the last 10,000 years, instead of the hundred-fold longer period of the million+ years of human evolution.”

Yes, but do you think it possible that life was making equations long before that? From the get go living things had the capacity to sense distinctions and, despite these distinctions, be able to say that one thing was like another, to equate them in some fashion. That’s the most fundamental cognitive mathematics and its precursors are likely electro-chemical in nature. Perhaps ultimately, somewhere in the mists of time, the universe began making distinctions and then, to pass the time, began to play with riffs of self-similarity.

Regardless of the modality of our modeling, I think we are existentially stuck with the fact that we can ride a bike or take it apart, but not do both simultaneously. There are reported to be states of apprehension where is all sensibly fits together into one glorious whole. Uncertain whether mathematics can take you there.

“We have to remember that what we observe is not nature herself, but nature exposed to our method of questioning.”